9.6 Notes – General Form

Each of the four types of equations we have studied so far this chapter; circle, parabola, ellipse, and hyperbola, can be described by slicing a double-right circular cone. The most general form of any conic can be written as: In this general form, the capital letters are just coefficients and our goal will be to transform any equation of this type into a standard form that we recognize.

ShapeGeneral Form:Standard FormPicture Parabola: horizontal Parabola: vertical Features of graph: Clues for Identifying Conic Types: A = 0 Vertex: (h, k) Only y squared B = 0 Only x squared Additional Points:2d from focus a > 0 open up or right, a < 0 opens down or left

ShapeGeneral FormStandard FormPicture Circle Features of graph: Clues for Identifying Conic Types: A = B (same denominator) Center: (h, k) r = Radius

ShapeGeneral Form:Standard FormPicture Ellipse: horizontal Ellipse: vertical Features of graph: Clues for Identifying Conic Types: A  B Center: (h, k) Both positive a is always the biggest denominator A  B Both positive Foci: c 2 = a 2 – b 2

ShapeGeneral Form:Standard FormPicture Hyperbola: horizontal Hyperbola : vertical Features of graph: Clues for Identifying Conic Types: A  B Center: (h, k) One negative a is always first denominator, make rectangle and asymptotes A  B One negative Foci: c 2 = a 2 + b 2

***Note: The vertex or center is always at __________. The h is with _____ and the k is with ________. When in parentheses, take the _____________. (h, k) xy opposite

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph Parabola

1. Graph 3 3 Opens:left Vertex:(1, 2) d =

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph Circle

Center: (1, –3) Radius: 2 2. Graph

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph Ellipse

3. Graph a = 5 b = 1 vertical Center:(2, 1)

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph hyperbola

4. Graph horizontal a = 5 b = 1 Center:(7, 2)

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph Parabola

5. Graph 2 2 Opens:up Vertex:(-2, 1) d =

Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph Ellipse

6. Graph a = 2 b = 1 horizontal Center:(5, 5)